The g in the subindex is "genus". A ball surface has genus zero, a donut surface has genus one, and so on, with the number of holes. The drawing is then a summation, the formula for each term given by the general rule above it.

The result of the summation would be, in this case, a "vacuum-to-vacuum" diagram, the probability for a string to come out of the vacuum, live for some time, and then disappear. It is the basis to build the same series for particular cases.